Differential equations

DIFFERENTIAL EQUATIONSORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS

GROUP MEMBERS:

•TAMOOR AHMED UET/SCET-13F-ME-004

•MALIK HAMMAD AZIZ UET/SCET-13F-ME-012

•Khawar RAZA UET/SCET-13F-ME-025

•M.ALI BHALLI ZADA UET/SCET-13F-ME-049

CONTENTS:• INVENTION OF DIFFERENTIAL EQUATION• INTRODUCTION TO DIFFERENTIAL EQUATION•TYPES OF DIFFERENTIAL EQUATION•BASIC CONCEPT OF ODE (ORDINARY

DIFFERENTIAL EQUATION)•TYPES OF ODE•BASIC CONCEPT OF PDE (PARTIAL DIFFERENTIAL

EQUATION)•TYPES OF PDE•APPLICATIONS OF DIFFERENTIAL EQUATIONS

INVENTION OF DIFFERENTIAL EQUATION:• In mathematics, the history of differential equations traces

the development of "differential equations" from calculus, which itself was independently invented by English physicist Isaac Newton and German mathematician Gottfried Leibniz. 

• The history of the subject of differential equations, in concise form, from a synopsis of the recent article “The History of Differential Equations, 1670-1950”

“Differential equations began with Leibniz, the Bernoulli brothers, and others from the 1680s, not long after Newton’s ‘fluxional equations’ in the 1670s.”

DIFFERENTIAL EQUATION:

• A Differential Equation is an equation containing the derivative of one or more dependent variables with respect to one or more independent variables.

• For Example,

TYPES OF DIFFERENTIAL EQUATION:

ODE (ORDINARY DIFFERENTIAL EQUATION): An equation contains only ordinary derivates of one

or more dependent variables of a single independent variable.

For Example, dy/dx + 5y = ex, (dx/dt) + (dy/dt) = 2x + y

PDE (PARTIAL DIFFERENTIAL EQUATION): An equation contains partial derivates of one or

more dependent variables of two or more independent variables.

For Example,

TYPES OF ODE:

FIRST ORDER ODE• FIRST ORDER LINEAR ODE• EXACT EQUATION• NON-LINEAR FIRST ORDER ODE• SEPERABLE EQUATION• BERNOULLI DIFFERENTIAL EQUATIONSECOND ORDER ODE• LINEAR SECOND ORDER ODE• HOMOGENEOUS SECOND ORDER ODE• INITIAL AND BOUNDARY VALUE PROBLEMS• NON-LINEAR SECOND ORDER ODE• NON-HOMOGENEOUS SECOND ORDER ODEHIGHER ORDER ODE• LINEAR NTH ORDER ODE• HOMOGENEOUS EQUATION• NON-HOMOGENEOUS EQUATION

FIRST ORDER ODE:

•A first order differential equation is an equation involving the unknown function y, its derivative y' and the variable x. We will only talk about explicit differential equations.

•General Form,

•For Example, 32 xdx

dy

• FIRST ORDER linear ODE:

•A first order linear differential equation has the following form:

•The general solution is given by

•Where

•called the integrating factor. If an initial condition is given, use it to find the constant C.

• EXACT EQUATION:

• Let a first order ordinary differential equation be expressible in this form:

M(x,y)+N(x,y)dy/dx=0 such that M and N are not homogeneous functions of the

same degree.

• However, suppose there happens to exist a function f(x,y) such that:

∂f/∂x=M, ∂f/∂y=N such that the second partial derivatives of f exist and are 

continuous.

• Then the expression Mdx+Ndy is called an exact differential, and the differential equation is called an exact differential equation.

SEPERABLEQUATION:

• A separable differential equation is any differential equation that we can write in the following form.

• Note that in order for a differential equation to be separable all the y's in the differential equation must be multiplied by the derivative and all the x's in the differential equation must be on the other side of the equal sign.

BERNOULLI EQUATION:

• where p(x) and q(x) are continuous functions on the interval we’re working on and n is a real number.  Differential equations in this form are called Bernoulli Equations.

• First notice that if   or   then the equation is linear and we already know how to solve it in these cases.  Therefore, in this section we’re going to be looking at solutions for values of n other than these two.

• In order to solve these we’ll first divide the differential equation by yA to get, • We are now going to use the substitution  to convert

this into a differential equation in terms of v.  As we’ll see this will lead to a differential equation that we can solve. 

SECOND ORDER ODE:

• The most general linear second order differential equation is in the form.

• In fact, we will rarely look at non-constant coefficient linear second order differential equations.  In the case where we assume constant coefficients we will use the following differential equation.

• Initially we will make our life easier by looking at differential equations with g(t) = 0.  When g(t) = 0 we call the Differential Equation Homogeneous and when   we call the Differential Equation Non-Homogeneous.

•So, let’s start thinking about how to go about solving a constant coefficient, homogeneous, linear, second order differential equation.  Here is the general constant coefficient, homogeneous, linear, second order differential equation.

•For Example,

INITIAL AND BOUNDARY VALUE PROBLEMS:• Boundary value problems are similar to initial value problems. A

boundary value problem has conditions specified at the extremes ("boundaries") of the independent variable in the equation whereas an initial value problem has all of the conditions specified at the same value of the independent variable (and that value is at the lower boundary of the domain, thus the term "initial" value).

• For example, if the independent variable is time over the domain [0,1], a boundary value problem would specify values for  at both  and , whereas an initial value problem would specify a value of    and    at time .

• Finding the temperature at all points of an iron bar with one end kept at absolute zero and the other end at the freezing point of water would be a boundary value problem.

• If the problem is dependent on both space and time, one could specify the value of the problem at a given point for all time the data or at a given time for all space.

• Concretely, an example of a boundary value (in one spatial dimension) is the problem

• to be solved for the unknown function   with the boundary conditions

• Without the boundary conditions, the general solution to this equation is

• From the boundary condition   one obtains

• which implies that   From the boundary condition   one finds

• and so   One sees that imposing boundary conditions allowed one to determine a unique solution, which in this case is

HIGHER ORDER ODE:

• The most general nth order linear differential equation is, nth – order linear differential equation with constant coefficients

•nth – order linear differential equation with variable coefficients,

xgyxadx

dyxa

dx

ydxa

dx

ydxa

dx

dyxa

n

n

nn

012

2

2

1

1 ......

xgyadx

dya

dx

yda

dx

yda

dx

yda

n

n

nn

n

n

012

2

21

1

1 ....

•For Example,

•The above equation is an example of Higher Order Homogeneous Differential ODE with initial conditions.

•Similarly, the above equation is an Higher Order Non-Homogeneous Differential ODE with coefficients.

APPLICATIONS OF ODE:

MODELLING WITH FIRST-ORDER EQUATIONS Newton’s Law of Cooling Electrical CircuitsMODELLING FREE MECHANICAL OSCILLATIONS No Damping Light Damping Heavy DampingMODELLING FORCED MECHANICAL

OSCILLATIONSCOMPUTER EXERCISE OR ACTIVITY

TYPES OF PDE(PARTIAL DIFFERENTIAL EQUATION):

LINEAR PDE

NON-LINEAR PDE

0322

032

032

PDENonlinear of Examples

0432

0)2cos(4312

:linear PDE of Example

sderivative its and function

unknown theinlinear isit iflinear is PDEA

2

ttxtxx

txtxx

ttxtxx

xtxx

xttxtxx

uuuu

uuu

uuu

uuu

tuuuu

Examples of PDE:

PDEs are used to model many systems in many different fields of science and engineering.

Important Examples:Laplace EquationHeat EquationWave Equation

LAPLACE EQUATION:

•Laplace Equation is used to describe the steady state distribution of heat in a body.

•Also used to describe the steady state distribution of electrical charge in a body.

0),,(),,(),,(

2

2

2

2

2

2

z

zyxu

y

zyxu

x

zyxu

HEAT EQUATION:

•The function u(x,y,z,t) is used to represent the temperature at time t in a physical body at a point with coordinates (x,y,z)

• is the thermal diffusivity. It is sufficient to consider the case = 1.

2

2

2

2

2

2

),,,(

z

u

y

u

x

u

t

tzyxu

WAVE EQUATION:

•The function u(x,y,z,t) is used to represent the displacement at time t of a particle whose position at rest is (x,y,z) .

•The constant c represents the propagation speed of the wave.

2

2

2

2

2

22

2

2

),,,(

z

u

y

u

x

uc

t

tzyxu